    Time-to-depth conversion and seismic velocity estimation using time-migration velocity  Next: Seismic Velocity Up: Cameron, Fomel, Sethian: Velocity Previous: Introduction

# Time Migration Velocity

Kirchhoff prestack time migration is commonly based on the following travel time approximation (Yilmaz, 2001). Let be a source, be a receiver, and be the reflection subsurface point. Then the total travel time from to and from to is approximated as (1)

where and are effective parameters of the subsurface point . The approximation usually takes the form of the double-square-root equation (2)

where and are the escape location and the travel time of the image ray (Hubral, 1977) from the subsurface point . Regarding this approximation, let us list four cases depending on the seismic velocity and the dimension of the problem:
2-D and 3-D, velocity is constant.
Equation 2 is exact, and .
2-D and 3-D, velocity depends only on the depth .
Equation 2 is a consequence of the truncated Taylor expansion for the travel time around the surface point . Velocity depends only on and is the root-mean-square velocity: (3)

In this case, the Dix inversion formula (Dix, 1955) is exact. We formally define the Dix velocity by inverting equation 3, as follows: (4)

2-D, velocity is arbitrary.
Equation 2 is a consequence of the truncated Taylor expansion for the travel time around the surface point . Velocity is a certain kind of mean velocity, and we establish its exact meaning in the next section.
3-D, velocity is arbitrary.
Equation 2 is heuristic and is not a consequence of the truncated Taylor expansion. In order to write an analog of travel time approximation 2 for 3-D, we use the relation (Hubral and Krey, 1980) (5)

where is the matrix of the second derivatives of the travel times from a subsurface point to the surface, is the matrix of radii of curvature of the emerging wave front from the point source , and is the velocity at the surface point . For convenience, we prefer to deal with matrix , which, according to equation 5 is (6)

The travel time approximation for 3-D implied by the Taylor expansion is   (7)  The entries of the matrix have dimension of squared velocity and can be chosen optimally in the process of time migration. It is possible to show, however, that one needs only the values of (8)

to perform the inversion. This means that the conventional 3-D prestack time migration with traveltime approximation 2 provides sufficient input for our inversion procedure in 3-D. The determinant in equation 8 is well approximated by the square of the Dix velocity obtained from the 3-D prestack time migration using the approximation given by equation 2.
One can employ more complex and accurate approximations than the double-square-root equations 2 and 7, i.e. the shifted hyperbola approximation (Siliqi and Bousquié, 2000). However, other known approximations also involve parameters equivalent to or .    Time-to-depth conversion and seismic velocity estimation using time-migration velocity  Next: Seismic Velocity Up: Cameron, Fomel, Sethian: Velocity Previous: Introduction

2013-03-02